Topicmef85039c8c75139e_1528449000663_0Topic

Points, lines and planes in space – doing exercises

Levelmef85039c8c75139e_1528449084556_0Level

Third

Core curriculummef85039c8c75139e_1528449076687_0Core curriculum

X. Stereometry

The basic level. The student:

1) identifies mutual position of lines in space, especially perpendicular lines that do not cross,

2) applies the concept of an angle between the line and the planeangle between the line and the planeangle between the line and the plane as well as the dihedral angledihedral angledihedral angle between half‑planes.

Timingmef85039c8c75139e_1528449068082_0Timing

45 minutes

General objectivemef85039c8c75139e_1528449523725_0General objective

Using mathematical objects, interpreting mathematical concepts.

Specific objectivesmef85039c8c75139e_1528449552113_0Specific objectives

1. Drawing lines in space, according to conditions from the task.

2. Doing exercises that use the concept of the angle between a line and a plane and the dihedral angledihedral angledihedral angle between half‑planes.

3. Communicating in English, developing basic mathematical, computer and scientific competences, developing learning skills.

Learning outcomesmef85039c8c75139e_1528450430307_0Learning outcomes

The student:

- Draws lines in space, according to conditions from the task.

- Does exercises that use the concept of the angle between a line and a plane and the dihedral angledihedral angledihedral angle between half‑planes.

Methodsmef85039c8c75139e_1528449534267_0Methods

1. Situational analysis

2. Task stations

Forms of workmef85039c8c75139e_1528449514617_0Forms of work

1. Individual work

2. Group work

Lesson stages

Introductionmef85039c8c75139e_1528450127855_0Introduction

Students discuss information about points, lines and planes they learnt during the last lesson. They watch posters created during the last lesson.

Proceduremef85039c8c75139e_1528446435040_0Procedure

Students work individually, using computers. They open the SLIDESHOW and observe drawings that illustrates relations between lines and planes in space.

[Slideshow]

The teacher divides students into four groups. Students in groups do exercises from each task stations. They get points for proper solutions.

The group that gets most points, gets marks from activity.

Station I - Mutual position of a line and a plane.
There is a parallelogram ABCD whose diagonals cross at point P and the point S that does not belong to the plane ABCD. Moreover, we know that line segments AS and SC as well as SD and SB are respectively equal to each other.

Define the location of the line segment PS with respect to the plane ABCD. Justify the answer.

Station II – Mutual position of lines in space.
Draw a cube ABCDEFGH. Fill in the table by writing four examples of proper lines, determined by vertices of this cube.

Station III - Angles between a line and a plane
There is a line segment AB located on the plane p and a point C located in a distance equal to the length of the line segment AB. Find the angle between the line AC and the plane p if the midpoint of the line segment AB is an orthographic projection of the point C on the plane p.

Station IV - The dihedral angledihedral angledihedral angle between two half‑planes.
Identify the angle of inclination of the side wall to the plane of the base in the drawn pyramid, knowing that all its edges have the same length.

[Illustration 1]

The teacher evaluates students’ work and clarifies doubts.

An extra task:
The line segment whose length is a is inclined to the plane p at the angle 60°. Calculate the length of the orthographic projection of this line segment on the plane p.

Lesson summarymef85039c8c75139e_1528450119332_0Lesson summary

Students do the revision exercises. Then together they sum‑up the classes, by formulating the conclusions to memorise.

- A line can be located on the plane, break the plane or have no common points with the plane.mef85039c8c75139e_1527752263647_0A line can be located on the plane, break the plane or have no common points with the plane.
- Two lines in space can overlap, be located on one plane or not be located on one plane and have no common points (be oblique).mef85039c8c75139e_1527752256679_0Two lines in space can overlap, be located on one plane or not be located on one plane and have no common points (be oblique).
- The line k and the plane p are perpendicular only and only if the line k is parallel to each line located on the plane p,
- The angle between the line k and the plane p is the acute angle between this line and orthographic projection on the plane p,
- If the line l breaks the plane p and is not perpendicular to it, the line k is an orthographic projection of the line l on the plane p, the line m is located on the plane p and crosses the line l, then the line m is perpendicular to the line l only and only if it is perpendicular to the line k (theorem about three perpendicular lines).
- The dihedral angledihedral angledihedral angle is the sum of two half‑planes with common edge and one of two areas that this half‑planes cut from the space,
- The linear anglelinear anglelinear angle of the dihedral angledihedral angledihedral angle is the common part of the dihedral angledihedral angledihedral angle and the plane perpendicular to its edge.

Selected words and expressions used in the lesson plan

angle between the line and the planeangle between the line and the planeangle between the line and the plane,

dihedral angledihedral angledihedral angle,

line breaking the planeline breaking the planeline breaking the plane,

line perpendicular to the planeline perpendicular to the planeline perpendicular to the plane,

linear anglelinear anglelinear angle,

location of a line and a planelocation of a line and a planelocation of a line and a plane,

location of two lines in spacelocation of two lines in spacelocation of two lines in space